Abstract
In this paper we analyse some bootstrap techniques to make inference in INAR(p) models. First of all, via Monte Carlo experiments we compare the performances of these methods when estimating the thinning parameters in INAR(p) models; we state the superiority of model-based INAR bootstrap approaches on block bootstrap in terms of low bias and Mean Square Error. Then we adopt the model-based bootstrap methods to obtain coherent predictions and confidence intervals in order to avoid difficulty in deriving the distributional properties. Finally, we present an empirical application.
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Notes
It is worth citing the work of McCabe et al. (2011) which estimate the forecast distribution non-parametrically within the context of the integer auto-regressive class of models and derive efficient probabilistic forecasts. To assess sampling variation in the full estimated forecast distribution, the authors use a subsampling method and prove its validity. Their approach is similar to that of Politis et al. (1999) and it is different from that we propose.
For the other several variants of block bootstrap and further details, see Kreiss and Lahiri (2012) and the reference therein.
As in the case of the IID bootstrap, the BB sample size is typically chosen to be of the same order as the original sample size. If b is not integer but denotes the smallest integer such that \(b l \ge n\), then one may select b blocks to generate the BB samples, and use only the first n values (Lahiri 2003, p. 26).
Our approach differs from Cardinal et al. (1999) and Kim and Park (2008) approach in the computation of the residuals. In particular, they compute the residuals as:
$$\begin{aligned} \hat{\varepsilon }_t = x_t - \sum _{i=1}^{p} \hat{\alpha }_i x_{t-i} \end{aligned}$$for \(t=p+1, \ldots , n,\) where \(\sum _{i=1}^{p} \hat{\alpha }_i x_{t-i}\) is the estimated conditional expectation of \(X_t,\) then consider the modified residuals defined by \(\tilde{\varepsilon }_t=[\hat{\varepsilon }_t]\) where \([\cdot ]\) represents the value rounded to the nearest integer.
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Bisaglia, L., Gerolimetto, M. Model-based INAR bootstrap for forecasting INAR(p) models. Comput Stat 34, 1815–1848 (2019). https://doi.org/10.1007/s00180-019-00902-1
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DOI: https://doi.org/10.1007/s00180-019-00902-1